×

Positive radial solutions of a quasilinear problem in an exterior domain with vanishing boundary conditions. (English) Zbl 1479.35424

Summary: In this work, we study the existence and nonexistence of positive radial solutions for the quasilinear equation \(\mathrm{div} (A(|\nabla u|)\nabla u)+\lambda k(|x|)f(u)=0\) in the exterior of a ball with vanishing boundary conditions using an approach based on a fixed point theorem for operators on Banach Space.

MSC:

35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B09 Positive solutions to PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Bandle, C.V. Coffman and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987), 322-345. · Zbl 0618.35043
[2] C. Bandle and M.K. Kwong, Semilinear elliptic problems in annular domains, J. Appl. Math. Phys. (ZAMP) 40 (1989), 245-257. · Zbl 0687.35036
[3] M. Chhetri, P. Drabek and R. Shivaji, Analysis of positive solutions for classes of quasilinear singular problems on exterior domains, Adv. Nonlinear Anal. 6 (2017), 447-459. · Zbl 1377.35120
[4] M. Chhetri, P. Drabek and R. Shivaji, S-shaped bifurcation diagrams in exterior domains, Positivity 23 (2019), 1147-1164. · Zbl 1428.35134
[5] S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), 211-221.
[6] C.V. Coffman and M. Marcus, Existenceand uniqueness results for semilinear Dirichlet problems in annuli, Arch. Rational Mech. Anal. 108 (1989), 293-307. · Zbl 0699.35092
[7] K. Deimling, Nonlinear Functional Analysis, Springer, 1995. · Zbl 0839.45005
[8] R. Dhanya, Q. Morris and R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl. 434 (2016), 1533-1548. · Zbl 1328.35024
[9] J.M. do O, S. Lorca, J. Sanchez and P. Ubilla, Non-homogeneous elliptic equations in exterior domain, Proc. Roy. Soc. Edinburgh Ser. A 136 (2006), 139-147. · Zbl 1281.35041
[10] I.M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Tansl. Ser. 2 29 (1963), 295-381. · Zbl 0127.04901
[11] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 1988. · Zbl 0661.47045
[12] D.D. Hai and R. Shivaji, Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl. 456 (2017), 872-881. · Zbl 1392.35110
[13] D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241-269. · Zbl 0266.34021
[14] L. Kong and J. Wang, Multiple positive solutions for the one-dimensional \(p\)-Laplacian, Nonlinear Anal. 42 (2000), 1327-1333. · Zbl 0961.34012
[15] M.A. Krasnosel’skiı, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. · Zbl 0121.10604
[16] S.S. Lin, On the existence of positive radial solutions for semilinear elliptic equations in annular domains, J. Differential Equations 81 (1989), 221-233. · Zbl 0691.35036
[17] W.M. Ni and J. Serrin, Non-existence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math. 39 (1986), 379-399. · Zbl 0602.35031
[18] W.M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo Suppl. 5 (1986), 171-185. · Zbl 0625.35028
[19] J. Sanchez, Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional \(p\)-laplacian, J. Math. Anal. Appl. 292 (2004), 401-414. · Zbl 1057.34012
[20] R. Stanczy, Positive solutions for superlinear elliptic equations, J. Math. Anal. Appl. 283 (2003), 159-166. · Zbl 1093.35027
[21] H. Wang, On the existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 109 (1994), 1-7. · Zbl 0798.34030
[22] J. Wang, The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 2275-2283. · Zbl 0884.34032
[23] H.Wang, On the structure of positive radial solutions for quasilinear equations in annular domains, Adv. Differential Equations 8 (2003), 111-128. · Zbl 1042.34052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.